Abstract

Tunneling ionization and harmonic generation in two-color fields were studied with a fundamental beam (ω) and its harmonics (2ω, 3ω), which were generated by a 100-fs Ti:sapphire laser. Ion yields of atoms and molecules were successfully controlled by means of a change in the relative phase between ω and 3ω pulses. Two-color interference was clearly observed in photoelectron spectra and harmonic spectra. In the ω–2ω field even-order harmonics were observed in which the intensity was almost equal to that of the odd harmonics because of an asymmetric optical field. These results were compared with the quasi-static model for ionization and with the quantum theory for harmonic generation.

© 1996 Optical Society of America

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  1. S. Augst, D. Strickland, D. D. Meyerhofer, S. L. Chin, and J. H. Eberly, Phys. Rev. Lett. 63, 2212 (1989); S. Augst, D. D. Meyerhofer, D. Strickland, and S. L. Chin, J. Opt. Soc. Am. B 8, 858 (1991).
    [CrossRef] [PubMed]
  2. V. Ammosov, N. B. Delone, and V. P. Krainov, Sov. Phys. JETP 64, 1191 (1986).
  3. P. B. Corkum, Phys. Rev. Lett. 71, 1994 (1993).
    [CrossRef] [PubMed]
  4. S. Watanabe, K. Kondo, Y. Nabekawa, A. Sagisaka, and Y. Kobayashi, Phys. Rev. Lett. 73, 2692 (1994).
    [CrossRef] [PubMed]
  5. D. W. Schumacher, F. Weihe, H. G. Muller, and P. H. Buchsbaum, Phys. Rev. Lett. 73, 1344 (1994).
    [CrossRef] [PubMed]
  6. H. G. Muller, P. H. Bucksbaum, D. W. Schumacher, and A. Zavriyev, J. Phys. B 23, 2761 (1990).
    [CrossRef]
  7. Y. Yian, C. Chen, and D. S. Elliott, Phys. Rev. Lett. 69, 2353 (1992).
    [CrossRef]
  8. C. Chen and D. S. Elliott, Phys. Rev. Lett. 65, 1737 (1990).
    [CrossRef] [PubMed]
  9. A. Szoke, K. C. Kulander, and J. N. Bardsley, J. Phys. B 24, 3165 (1991).
    [CrossRef]
  10. K. J. Schafer and K. C. Kulander, Phys. Rev. A 45, 8026 (1991).
    [CrossRef]
  11. R. M. Potvliege and H. G. Smith, J. Phys. B 25, 2501 (1992).
    [CrossRef]
  12. R. M. Potlviege and P. H. G. Smith, Phys. Rev. A 49, 3110 (1994).
    [CrossRef]
  13. P. B. Corkum, N. H. Burnett, and F. Brunel, Phys. Rev. Lett. 62, 1259 (1989).
    [CrossRef] [PubMed]
  14. K. J. Schafer, B. Young, L. F. DiMauro, and K. C. Kulander, Phy. Rev. Lett. 70, 1599 (1993).
    [CrossRef]
  15. M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L’Huillier, and P. B. Corkum, Phys. Rev. A 49, 2117 (1994).
    [CrossRef] [PubMed]
  16. D. T. Strickland, Y. Beaudoin, P. Dietrich, and P. B. Corkum, Phys. Rev. Lett. 68, 2755 (1992).
    [CrossRef] [PubMed]
  17. Y. Nabekawa, K. Kondo, N. Sarukura, K. Sajiki, and S. Watanabe, Opt. Lett. 18, 1922 (1993).
    [CrossRef] [PubMed]
  18. M. D. Perry and J. K. Crane, Phys. Rev. A 48, R4051 (1993).
    [CrossRef]
  19. W. Becker, S. Long, and J. K. McIver, Phys. Rev. A 41, 4112 (1990).
    [CrossRef] [PubMed]
  20. L. A. Lompre, A. L’Huillier, M. Ferray, P. Monot, G. Mainfray, and C. Manus, J. Opt. Soc. Am. B 7, 754 (1990).
    [CrossRef]

1994 (4)

S. Watanabe, K. Kondo, Y. Nabekawa, A. Sagisaka, and Y. Kobayashi, Phys. Rev. Lett. 73, 2692 (1994).
[CrossRef] [PubMed]

D. W. Schumacher, F. Weihe, H. G. Muller, and P. H. Buchsbaum, Phys. Rev. Lett. 73, 1344 (1994).
[CrossRef] [PubMed]

R. M. Potlviege and P. H. G. Smith, Phys. Rev. A 49, 3110 (1994).
[CrossRef]

M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L’Huillier, and P. B. Corkum, Phys. Rev. A 49, 2117 (1994).
[CrossRef] [PubMed]

1993 (4)

K. J. Schafer, B. Young, L. F. DiMauro, and K. C. Kulander, Phy. Rev. Lett. 70, 1599 (1993).
[CrossRef]

P. B. Corkum, Phys. Rev. Lett. 71, 1994 (1993).
[CrossRef] [PubMed]

Y. Nabekawa, K. Kondo, N. Sarukura, K. Sajiki, and S. Watanabe, Opt. Lett. 18, 1922 (1993).
[CrossRef] [PubMed]

M. D. Perry and J. K. Crane, Phys. Rev. A 48, R4051 (1993).
[CrossRef]

1992 (3)

R. M. Potvliege and H. G. Smith, J. Phys. B 25, 2501 (1992).
[CrossRef]

Y. Yian, C. Chen, and D. S. Elliott, Phys. Rev. Lett. 69, 2353 (1992).
[CrossRef]

D. T. Strickland, Y. Beaudoin, P. Dietrich, and P. B. Corkum, Phys. Rev. Lett. 68, 2755 (1992).
[CrossRef] [PubMed]

1991 (2)

A. Szoke, K. C. Kulander, and J. N. Bardsley, J. Phys. B 24, 3165 (1991).
[CrossRef]

K. J. Schafer and K. C. Kulander, Phys. Rev. A 45, 8026 (1991).
[CrossRef]

1990 (4)

C. Chen and D. S. Elliott, Phys. Rev. Lett. 65, 1737 (1990).
[CrossRef] [PubMed]

H. G. Muller, P. H. Bucksbaum, D. W. Schumacher, and A. Zavriyev, J. Phys. B 23, 2761 (1990).
[CrossRef]

W. Becker, S. Long, and J. K. McIver, Phys. Rev. A 41, 4112 (1990).
[CrossRef] [PubMed]

L. A. Lompre, A. L’Huillier, M. Ferray, P. Monot, G. Mainfray, and C. Manus, J. Opt. Soc. Am. B 7, 754 (1990).
[CrossRef]

1989 (2)

P. B. Corkum, N. H. Burnett, and F. Brunel, Phys. Rev. Lett. 62, 1259 (1989).
[CrossRef] [PubMed]

S. Augst, D. Strickland, D. D. Meyerhofer, S. L. Chin, and J. H. Eberly, Phys. Rev. Lett. 63, 2212 (1989); S. Augst, D. D. Meyerhofer, D. Strickland, and S. L. Chin, J. Opt. Soc. Am. B 8, 858 (1991).
[CrossRef] [PubMed]

1986 (1)

V. Ammosov, N. B. Delone, and V. P. Krainov, Sov. Phys. JETP 64, 1191 (1986).

Ammosov, V.

V. Ammosov, N. B. Delone, and V. P. Krainov, Sov. Phys. JETP 64, 1191 (1986).

Augst, S.

S. Augst, D. Strickland, D. D. Meyerhofer, S. L. Chin, and J. H. Eberly, Phys. Rev. Lett. 63, 2212 (1989); S. Augst, D. D. Meyerhofer, D. Strickland, and S. L. Chin, J. Opt. Soc. Am. B 8, 858 (1991).
[CrossRef] [PubMed]

Balcou, Ph.

M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L’Huillier, and P. B. Corkum, Phys. Rev. A 49, 2117 (1994).
[CrossRef] [PubMed]

Bardsley, J. N.

A. Szoke, K. C. Kulander, and J. N. Bardsley, J. Phys. B 24, 3165 (1991).
[CrossRef]

Beaudoin, Y.

D. T. Strickland, Y. Beaudoin, P. Dietrich, and P. B. Corkum, Phys. Rev. Lett. 68, 2755 (1992).
[CrossRef] [PubMed]

Becker, W.

W. Becker, S. Long, and J. K. McIver, Phys. Rev. A 41, 4112 (1990).
[CrossRef] [PubMed]

Brunel, F.

P. B. Corkum, N. H. Burnett, and F. Brunel, Phys. Rev. Lett. 62, 1259 (1989).
[CrossRef] [PubMed]

Buchsbaum, P. H.

D. W. Schumacher, F. Weihe, H. G. Muller, and P. H. Buchsbaum, Phys. Rev. Lett. 73, 1344 (1994).
[CrossRef] [PubMed]

Bucksbaum, P. H.

H. G. Muller, P. H. Bucksbaum, D. W. Schumacher, and A. Zavriyev, J. Phys. B 23, 2761 (1990).
[CrossRef]

Burnett, N. H.

P. B. Corkum, N. H. Burnett, and F. Brunel, Phys. Rev. Lett. 62, 1259 (1989).
[CrossRef] [PubMed]

Chen, C.

Y. Yian, C. Chen, and D. S. Elliott, Phys. Rev. Lett. 69, 2353 (1992).
[CrossRef]

C. Chen and D. S. Elliott, Phys. Rev. Lett. 65, 1737 (1990).
[CrossRef] [PubMed]

Chin, S. L.

S. Augst, D. Strickland, D. D. Meyerhofer, S. L. Chin, and J. H. Eberly, Phys. Rev. Lett. 63, 2212 (1989); S. Augst, D. D. Meyerhofer, D. Strickland, and S. L. Chin, J. Opt. Soc. Am. B 8, 858 (1991).
[CrossRef] [PubMed]

Corkum, P. B.

M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L’Huillier, and P. B. Corkum, Phys. Rev. A 49, 2117 (1994).
[CrossRef] [PubMed]

P. B. Corkum, Phys. Rev. Lett. 71, 1994 (1993).
[CrossRef] [PubMed]

D. T. Strickland, Y. Beaudoin, P. Dietrich, and P. B. Corkum, Phys. Rev. Lett. 68, 2755 (1992).
[CrossRef] [PubMed]

P. B. Corkum, N. H. Burnett, and F. Brunel, Phys. Rev. Lett. 62, 1259 (1989).
[CrossRef] [PubMed]

Crane, J. K.

M. D. Perry and J. K. Crane, Phys. Rev. A 48, R4051 (1993).
[CrossRef]

Delone, N. B.

V. Ammosov, N. B. Delone, and V. P. Krainov, Sov. Phys. JETP 64, 1191 (1986).

Dietrich, P.

D. T. Strickland, Y. Beaudoin, P. Dietrich, and P. B. Corkum, Phys. Rev. Lett. 68, 2755 (1992).
[CrossRef] [PubMed]

DiMauro, L. F.

K. J. Schafer, B. Young, L. F. DiMauro, and K. C. Kulander, Phy. Rev. Lett. 70, 1599 (1993).
[CrossRef]

Eberly, J. H.

S. Augst, D. Strickland, D. D. Meyerhofer, S. L. Chin, and J. H. Eberly, Phys. Rev. Lett. 63, 2212 (1989); S. Augst, D. D. Meyerhofer, D. Strickland, and S. L. Chin, J. Opt. Soc. Am. B 8, 858 (1991).
[CrossRef] [PubMed]

Elliott, D. S.

Y. Yian, C. Chen, and D. S. Elliott, Phys. Rev. Lett. 69, 2353 (1992).
[CrossRef]

C. Chen and D. S. Elliott, Phys. Rev. Lett. 65, 1737 (1990).
[CrossRef] [PubMed]

Ferray, M.

Ivanov, M. Yu.

M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L’Huillier, and P. B. Corkum, Phys. Rev. A 49, 2117 (1994).
[CrossRef] [PubMed]

Kobayashi, Y.

S. Watanabe, K. Kondo, Y. Nabekawa, A. Sagisaka, and Y. Kobayashi, Phys. Rev. Lett. 73, 2692 (1994).
[CrossRef] [PubMed]

Kondo, K.

S. Watanabe, K. Kondo, Y. Nabekawa, A. Sagisaka, and Y. Kobayashi, Phys. Rev. Lett. 73, 2692 (1994).
[CrossRef] [PubMed]

Y. Nabekawa, K. Kondo, N. Sarukura, K. Sajiki, and S. Watanabe, Opt. Lett. 18, 1922 (1993).
[CrossRef] [PubMed]

Krainov, V. P.

V. Ammosov, N. B. Delone, and V. P. Krainov, Sov. Phys. JETP 64, 1191 (1986).

Kulander, K. C.

K. J. Schafer, B. Young, L. F. DiMauro, and K. C. Kulander, Phy. Rev. Lett. 70, 1599 (1993).
[CrossRef]

A. Szoke, K. C. Kulander, and J. N. Bardsley, J. Phys. B 24, 3165 (1991).
[CrossRef]

K. J. Schafer and K. C. Kulander, Phys. Rev. A 45, 8026 (1991).
[CrossRef]

L’Huillier, A.

M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L’Huillier, and P. B. Corkum, Phys. Rev. A 49, 2117 (1994).
[CrossRef] [PubMed]

L. A. Lompre, A. L’Huillier, M. Ferray, P. Monot, G. Mainfray, and C. Manus, J. Opt. Soc. Am. B 7, 754 (1990).
[CrossRef]

Lewenstein, M.

M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L’Huillier, and P. B. Corkum, Phys. Rev. A 49, 2117 (1994).
[CrossRef] [PubMed]

Lompre, L. A.

Long, S.

W. Becker, S. Long, and J. K. McIver, Phys. Rev. A 41, 4112 (1990).
[CrossRef] [PubMed]

Mainfray, G.

Manus, C.

McIver, J. K.

W. Becker, S. Long, and J. K. McIver, Phys. Rev. A 41, 4112 (1990).
[CrossRef] [PubMed]

Meyerhofer, D. D.

S. Augst, D. Strickland, D. D. Meyerhofer, S. L. Chin, and J. H. Eberly, Phys. Rev. Lett. 63, 2212 (1989); S. Augst, D. D. Meyerhofer, D. Strickland, and S. L. Chin, J. Opt. Soc. Am. B 8, 858 (1991).
[CrossRef] [PubMed]

Monot, P.

Muller, H. G.

D. W. Schumacher, F. Weihe, H. G. Muller, and P. H. Buchsbaum, Phys. Rev. Lett. 73, 1344 (1994).
[CrossRef] [PubMed]

H. G. Muller, P. H. Bucksbaum, D. W. Schumacher, and A. Zavriyev, J. Phys. B 23, 2761 (1990).
[CrossRef]

Nabekawa, Y.

S. Watanabe, K. Kondo, Y. Nabekawa, A. Sagisaka, and Y. Kobayashi, Phys. Rev. Lett. 73, 2692 (1994).
[CrossRef] [PubMed]

Y. Nabekawa, K. Kondo, N. Sarukura, K. Sajiki, and S. Watanabe, Opt. Lett. 18, 1922 (1993).
[CrossRef] [PubMed]

Perry, M. D.

M. D. Perry and J. K. Crane, Phys. Rev. A 48, R4051 (1993).
[CrossRef]

Potlviege, R. M.

R. M. Potlviege and P. H. G. Smith, Phys. Rev. A 49, 3110 (1994).
[CrossRef]

Potvliege, R. M.

R. M. Potvliege and H. G. Smith, J. Phys. B 25, 2501 (1992).
[CrossRef]

Sagisaka, A.

S. Watanabe, K. Kondo, Y. Nabekawa, A. Sagisaka, and Y. Kobayashi, Phys. Rev. Lett. 73, 2692 (1994).
[CrossRef] [PubMed]

Sajiki, K.

Sarukura, N.

Schafer, K. J.

K. J. Schafer, B. Young, L. F. DiMauro, and K. C. Kulander, Phy. Rev. Lett. 70, 1599 (1993).
[CrossRef]

K. J. Schafer and K. C. Kulander, Phys. Rev. A 45, 8026 (1991).
[CrossRef]

Schumacher, D. W.

D. W. Schumacher, F. Weihe, H. G. Muller, and P. H. Buchsbaum, Phys. Rev. Lett. 73, 1344 (1994).
[CrossRef] [PubMed]

H. G. Muller, P. H. Bucksbaum, D. W. Schumacher, and A. Zavriyev, J. Phys. B 23, 2761 (1990).
[CrossRef]

Smith, H. G.

R. M. Potvliege and H. G. Smith, J. Phys. B 25, 2501 (1992).
[CrossRef]

Smith, P. H. G.

R. M. Potlviege and P. H. G. Smith, Phys. Rev. A 49, 3110 (1994).
[CrossRef]

Strickland, D.

S. Augst, D. Strickland, D. D. Meyerhofer, S. L. Chin, and J. H. Eberly, Phys. Rev. Lett. 63, 2212 (1989); S. Augst, D. D. Meyerhofer, D. Strickland, and S. L. Chin, J. Opt. Soc. Am. B 8, 858 (1991).
[CrossRef] [PubMed]

Strickland, D. T.

D. T. Strickland, Y. Beaudoin, P. Dietrich, and P. B. Corkum, Phys. Rev. Lett. 68, 2755 (1992).
[CrossRef] [PubMed]

Szoke, A.

A. Szoke, K. C. Kulander, and J. N. Bardsley, J. Phys. B 24, 3165 (1991).
[CrossRef]

Watanabe, S.

S. Watanabe, K. Kondo, Y. Nabekawa, A. Sagisaka, and Y. Kobayashi, Phys. Rev. Lett. 73, 2692 (1994).
[CrossRef] [PubMed]

Y. Nabekawa, K. Kondo, N. Sarukura, K. Sajiki, and S. Watanabe, Opt. Lett. 18, 1922 (1993).
[CrossRef] [PubMed]

Weihe, F.

D. W. Schumacher, F. Weihe, H. G. Muller, and P. H. Buchsbaum, Phys. Rev. Lett. 73, 1344 (1994).
[CrossRef] [PubMed]

Yian, Y.

Y. Yian, C. Chen, and D. S. Elliott, Phys. Rev. Lett. 69, 2353 (1992).
[CrossRef]

Young, B.

K. J. Schafer, B. Young, L. F. DiMauro, and K. C. Kulander, Phy. Rev. Lett. 70, 1599 (1993).
[CrossRef]

Zavriyev, A.

H. G. Muller, P. H. Bucksbaum, D. W. Schumacher, and A. Zavriyev, J. Phys. B 23, 2761 (1990).
[CrossRef]

J. Opt. Soc. Am. B (1)

J. Phys. B (3)

R. M. Potvliege and H. G. Smith, J. Phys. B 25, 2501 (1992).
[CrossRef]

H. G. Muller, P. H. Bucksbaum, D. W. Schumacher, and A. Zavriyev, J. Phys. B 23, 2761 (1990).
[CrossRef]

A. Szoke, K. C. Kulander, and J. N. Bardsley, J. Phys. B 24, 3165 (1991).
[CrossRef]

Opt. Lett. (1)

Phy. Rev. Lett. (1)

K. J. Schafer, B. Young, L. F. DiMauro, and K. C. Kulander, Phy. Rev. Lett. 70, 1599 (1993).
[CrossRef]

Phys. Rev. A (5)

M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L’Huillier, and P. B. Corkum, Phys. Rev. A 49, 2117 (1994).
[CrossRef] [PubMed]

R. M. Potlviege and P. H. G. Smith, Phys. Rev. A 49, 3110 (1994).
[CrossRef]

M. D. Perry and J. K. Crane, Phys. Rev. A 48, R4051 (1993).
[CrossRef]

W. Becker, S. Long, and J. K. McIver, Phys. Rev. A 41, 4112 (1990).
[CrossRef] [PubMed]

K. J. Schafer and K. C. Kulander, Phys. Rev. A 45, 8026 (1991).
[CrossRef]

Phys. Rev. Lett. (8)

Y. Yian, C. Chen, and D. S. Elliott, Phys. Rev. Lett. 69, 2353 (1992).
[CrossRef]

C. Chen and D. S. Elliott, Phys. Rev. Lett. 65, 1737 (1990).
[CrossRef] [PubMed]

S. Augst, D. Strickland, D. D. Meyerhofer, S. L. Chin, and J. H. Eberly, Phys. Rev. Lett. 63, 2212 (1989); S. Augst, D. D. Meyerhofer, D. Strickland, and S. L. Chin, J. Opt. Soc. Am. B 8, 858 (1991).
[CrossRef] [PubMed]

P. B. Corkum, Phys. Rev. Lett. 71, 1994 (1993).
[CrossRef] [PubMed]

S. Watanabe, K. Kondo, Y. Nabekawa, A. Sagisaka, and Y. Kobayashi, Phys. Rev. Lett. 73, 2692 (1994).
[CrossRef] [PubMed]

D. W. Schumacher, F. Weihe, H. G. Muller, and P. H. Buchsbaum, Phys. Rev. Lett. 73, 1344 (1994).
[CrossRef] [PubMed]

P. B. Corkum, N. H. Burnett, and F. Brunel, Phys. Rev. Lett. 62, 1259 (1989).
[CrossRef] [PubMed]

D. T. Strickland, Y. Beaudoin, P. Dietrich, and P. B. Corkum, Phys. Rev. Lett. 68, 2755 (1992).
[CrossRef] [PubMed]

Sov. Phys. JETP (1)

V. Ammosov, N. B. Delone, and V. P. Krainov, Sov. Phys. JETP 64, 1191 (1986).

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Figures (5)

Fig. 1
Fig. 1

Ion yield of H+ versus relative phase between ω and 3ω. The relative phase (ϕ) is defined by Eω sin (ωt + ϕ) + E3ω sin(3ωt). The intensity of ω and the field ratio (E3ω/Eω) are 4 × 1014 W/cm2 and 0.5, respectively. The solid curve is calculated with the quasi-static model.

Fig. 2
Fig. 2

Photoelectron spectra in Ar, obtained with ω, 3ω, and two colors (ω + 3ω). The intensity of ω and the field ratio are 3 × 1014 W/cm2 and 0.4. (a) Photoelectron number versus photoelectron energy. (b) Time-of-flight spectrum of only the 3ω field. Clear ATI peaks (1, 2, and 3) are observed.

Fig. 3
Fig. 3

Harmonic distribution in He obtained with ω (open circles) and two colors (filled circles, ω field and parallel-polarized 2ω field; +’s, ω field and orthogonally polarized 2ω field). The intensity of ω and field ratio are 1 × 1015 W/cm2 and 0.5.

Fig. 4
Fig. 4

Harmonic distributions in Ne calculated by quantum theory. The intensity of ω is 5 × 1014 W/cm2. The solid curve is the distribution for ω only. The dashed curves are distributions for the ω–3ω field, where the field ratio is 0.4. The relative phase differences are ρ = π/6 and 5π/6, respectively.

Fig. 5
Fig. 5

Harmonic distributions in He calculated by quantum theory. The intensity of ω is 5 × 1014 W/cm2. The filled circles connected by thick solid curves are for r = 0, the open squares connected by dotted curves are for r = 0.01 and ρ = 2π/3, the open triangles connected by thin solid curves are for r = 0.1 and ρ = 2π/3, and the open circles connected by thick solid curves are for r = 0.4 and ρ = 0.

Equations (27)

Equations on this page are rendered with MathJax. Learn more.

E ( t ) = E ω [ sin ( ω t ) + r sin ( k ω t + ρ ) ] ,
W av = 1 T 0 T W dc [ E ( t ) ] d t ,
ε ( t 0 ) = 2 U p [ cos ( ω t 0 ) + ( r / k ) cos ( k ω t 0 + ρ ) ] 2 .
E ( t ) = E ω [ cos ( ω t ) + r cos ( k ω t + σ ) ] ,
σ = ρ + ( k 1 ) ( π / 2 ) .
x 2 K + 1 = ( U p π α ) 3 / 2 2 α m 2 = m 3 = m 4 = ( i ) m 1 + m 2 + m 3 m 4 + 1 × 0 d τ ( π 1 / α + i τ / 2 ) 3 / 2 × exp { i [ ( I p K ) τ + F 1 ( τ ) + F 2 ( τ ) ] } × exp [ i ( 2 m 2 + m 3 + m 4 ) σ ] × [ L 1 ( τ ) + L 2 ( τ ) + LC ( τ ) ] ,
K = m 1 + km 2 + 1 + k 2 m 3 1 k 2 m 4 .
Ψ ( x ) = ( α π ) 3 / 4 exp ( α | x | 2 / 2 ) .
ϕ i ( t ; r , z ) = k i ( z + 2 zr 2 b i 2 + 4 z 2 ) + ρ i ω i t ,
B 1 ( τ ) = 2 sin 2 ( τ / 2 ) τ 2 + sin ( τ ) τ 1 2 ,
B 2 ( τ ) = 2 sin 2 ( k τ / 2 ) ( k τ ) 2 + sin ( k τ ) k τ 1 2 ,
D 1 ( τ ) = 2 B 1 ( τ ) 1 + cos ( τ ) ,
D 2 ( τ ) = 2 B 2 ( τ ) 1 + cos ( k τ ) ,
F 1 ( τ ) = U p [ τ 4 sin 2 ( τ / 2 ) τ + 4 i ( 1 / α ) B 1 ( τ ) ] ,
F 2 ( τ ) = 1 k ( r k ) 2 × U p [ k τ 4 sin 2 ( k τ / 2 ) k τ + 4 i ( k / α ) B 2 ( τ ) ] ,
C 1 ( τ ) = U p [ sin ( τ ) 4 sin 2 ( τ / 2 ) τ 2 i ( 1 / α ) D 1 ( τ ) ] ,
C 2 ( τ ) = 1 k ( r k ) 2 U p [ sin ( k τ ) 4 sin 2 ( k τ / 2 ) k τ 2 i ( k / α ) D 2 ( τ ) ] ,
δ ± ( τ ) = cos ( 1 ± k 2 τ ) + 4 k τ 2 sin ( τ / 2 ) sin ( k τ / 2 ) 2 [ 1 τ sin ( τ / 2 ) cos ( k τ / 2 ) + 1 k τ sin ( k τ / 2 ) cos ( τ / 2 ) ] ,
γ ± ( τ ) = 1 1 ± k sin ( 1 ± k 2 τ ) 1 k τ sin ( τ / 2 ) sin ( k τ / 2 ) ,
G ± ( τ ) = 4 ( r k ) U p [ γ ± ( τ ) i ( 1 / α ) δ ± ( τ ) ] .
m 1 ( τ ) = J m 1 [ C 1 ( τ ) ] ,
m 2 ( τ ) = J m 2 [ C 2 ( τ ) ] ,
m 3 ( τ ) = J m 3 [ G + ( τ ) ] ,
m 4 ( τ ) = J m 4 [ G ( τ ) ] .
L 1 ( τ ) = m 2 ( τ ) m 3 ( τ ) m 4 ( τ ) × { i m 1 + 1 ( τ ) [ { [ D 1 ( τ ) + ( r / k ) 2 D 2 ( τ ) ] + exp ( i τ ) B 1 ( τ ) } k ( r / k ) 2 ( exp { i [ ( 1 + k ) / 2 ] τ } δ ( τ ) exp { i [ ( 1 k ) / 2 ] τ } δ + ( τ ) ) ] + m 1 ( τ ) [ { exp ( i τ ) [ D 1 ( τ ) + ( r / k ) 2 D 2 ( τ ) ] + B 1 ( τ ) } k ( r / k ) 2 ( exp { i [ ( 1 k ) / 2 ] τ } δ ( τ ) exp { i [ ( 1 + k ) / 2 ] τ } δ + ( τ ) ) ] m 1 + 2 ( τ ) B 1 ( τ ) i m 1 1 ( τ ) exp ( i τ ) B 1 ( τ ) } ,
L 2 ( τ ) = i ( r / k ) m 1 ( τ ) m 2 ( τ ) × { m 3 + 1 ( τ ) m 4 ( τ ) × [ k ( exp { i [ ( 1 k ) / 2 ] τ } × [ D 1 ( τ ) + ( r / k ) 2 D 2 ( τ ) ] + ( r / k ) 2 × exp { i [ ( 1 + k ) / 2 ] τ } B 2 ( τ ) ) [ exp ( i τ ) δ ( τ ) δ + ( τ ) ] ] + m 3 ( τ ) m 4 1 ( τ ) [ k ( exp { i [ ( 1 + k ) / 2 ] τ } × [ D 1 ( τ ) + ( r / k ) 2 D 2 ( τ ) ] + ( r / k ) 2 × exp { i [ ( 1 k ) / 2 ] τ } B 2 ( τ ) ) [ δ ( τ ) exp ( i τ ) δ + ( τ ) ] ] } + k ( r / k ) 3 m 1 ( τ ) × ( m 2 + 1 ( τ ) m 3 + 1 ( τ ) m 4 ( τ ) × exp { i [ ( 1 k ) / 2 ] τ } B 2 ( τ ) + m 2 1 ( τ ) m 3 ( τ ) × m 4 1 ( τ ) exp { i [ ( 1 + k ) / 2 ] τ } B 2 ( τ ) )
LC ( τ ) = ( r / k ) m 2 ( τ ) × [ m 1 + 1 ( τ ) m 3 + 1 ( τ ) m 4 ( τ ) × ( k exp { i [ ( 1 k ) / 2 ] τ } × B 1 ( τ ) δ ( τ ) ) + i m 1 ( τ ) m 3 1 ( τ ) m 4 ( τ ) ( k exp { i [ ( 1 + k ) / 2 ] τ } × B 1 ( τ ) exp ( i τ ) δ ( τ ) ) + m 1 + 1 ( τ ) m 3 ( τ ) m 4 1 ( τ ) × ( k exp { i [ ( 1 + k ) / 2 ] τ } B 1 ( τ ) + δ + ( τ ) ) + i m 1 ( τ ) m 3 ( τ ) m 4 + 1 ( τ ) × ( k exp { i [ ( 1 k ) / 2 ] τ } × B 1 ( τ ) + exp ( i τ ) δ + ( τ ) ) ] + ( r / k ) 2 m 3 ( τ ) m 4 ( τ ) [ m 1 + 1 ( τ ) m 2 + 1 ( τ ) × ( B 2 ( τ ) k exp { i [ ( 1 k ) / 2 ] τ } δ ( τ ) ) i m 1 ( τ ) m 2 1 ( τ ) ( exp ( i τ ) × ( B 2 ( τ ) k exp { i [ ( 1 + k ) / 2 ] τ } δ ( τ ) ) + i m 1 ( τ ) m 2 + 1 ( τ ) ( exp ( i τ ) × ( B 2 ( τ ) + k exp { i [ ( 1 k ) / 2 ] τ } δ + ( τ ) ) + m 1 + 1 ( τ ) m 2 1 ( τ ) ( B 2 ( τ ) + k exp { i [ ( 1 + k ) / 2 ] τ } δ + ( τ ) ) ] .

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